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Geodesic deviation compares neighboring members of a family of freely falling trajectories. Subtracting their geodesic equations to first order in the connecting vector gives a covariant relative-acceleration equation governed by the Riemann curvature tensor. For a slowly moving satellite, $u^0=c\dot{t}\simeq c$ dominates the spatial components. Choose simultaneous spatial separation in a locally transported frame, take the weak-field approximation, and neglect terms quadratic in the small separation and higher velocity corrections. The leading spatial equation becomesReplacing the transported covariant derivatives by ordinary derivatives is a leading local or Newtonian approximation; a rotating orbital frame would introduce additional inertial terms. In the convention established above, the instantaneous tidal acceleration stretches radial separations and compresses transverse ones.

The growth expected for nearby different-radius circular orbits is especially transparent from Kepler's third law. If the radii are $R$ and $R+\delta R$, their coordinate angular frequencies differ by $\delta \Omega \simeq -(3/2)\Omega \delta R/R$. Initially aligned orbits therefore acquire an along-track separation $R\delta \Omega t\simeq -(3/2)\Omega \delta Rt$. Over the interval in which the nearby-orbit approximation remains valid, this produces a growing separation without needing to solve the deviation equation.

This growth is not universal or indefinitely unbounded. Equal-radius co-orbiting geodesics with a fixed phase offset have constant separation, and slightly inclined equal-radius circular orbits have bounded, periodically varying separation. Even different-radius circular orbits have bounded exact positions; their relative angle eventually wraps around. The secular drift statement assumes distinct radii and describes the local differential-orbit regime.

Take the real physical wave and write $h(t,z)=A\cos [k(z-ct)]$. The specified amplitude has zero trace, so its metric perturbation equals its trace-reversed metric perturbation. It is in transverse-traceless gauge:to first order. Assume the freely falling particles are initially at rest in these coordinates. Since $g_{00}=-1$ and $g_{0i}=0$ in coordinates $(ct,x,y,z)$,The geodesic equations therefore admitUniqueness preserves these initial conditions. Thus the coordinate locations remain fixed; this conclusion does not apply to arbitrary initially moving particles.

Coordinate position is not measured separation. In the central observer's local orthonormal frame in spacetime, and in the long-wavelength limit appropriate to a small detector, the transverse physical separations areto first order. In particular the two principal semiaxes of the ring areFor a point at angular position $\phi$ the radial displacement isThe freely falling ring in transverse-traceless gauge alternates between ellipses elongated in the $x$ and $y$ directions. Its area is unchanged to first order. This is the plus gravitational wave polarization; the axes exchange their roles after half a period.

An observer at the center detects these changing distances, for example by exchanging light signals with the particles. Equivalently geodesic deviation gives the local tidal accelerationto the same order. There is physical relative motion and nonzero Riemann curvature tensor, even though the selected coordinate positions are constant. For a detector comparable to a wavelength, its light-travel-time response must be included rather than treating all distances as simultaneously measured local lengths.

The geodesic deviation equation compares neighboring geodesics by covariant derivatives. Replacing those derivatives by ordinary derivatives of coordinate components requires a parallel transport frame along the reference geodesic. There is also an affine parameter normalization to specify: the argument of $M$ is $u$, and it becomes $\tau$ only after choosing $u=\tau$.

For the intended local tidal calculation, take the reference timelike geodesicHere $g(V,V)=-1$, so $\tau$ is proper time. Both $F$ and its first partial derivatives vanish on $\gamma$. All Christoffel symbols therefore vanish along this curve, and its coordinate basis is parallel transport along it. With the Riemann curvature tensor from part (b),The Jacobi field equation thus becomes, in this adapted frame,A purely transverse initial Jacobi field, with ${\sigma }^u={\sigma }^v={\dot{\sigma }}^u={\dot{\sigma }}^v=0$ initially, remains transverse. This transverse tidal equation of a Brinkmann plane wave describes the relative tidal acceleration of the freely falling bodies.

In vacuum write the symmetric matrix asIts instantaneous eigenvalues are $\pm \sqrt{A^2+B^2}$. The tidal acceleration stretches along one principal axis and squeezes along the perpendicular axis, with the axes allowed to vary with $u$. This describes accelerations; it does not imply monotonically growing and shrinking separations for arbitrary initial velocities. For the simple constant profile $A={\omega }^2$, $B=0$,Thus the geodesic deviation can be defocusing in one direction and oscillatory in the other. For a varying profile, diagonalizing at each instant does not decouple the ordinary differential equations without also including derivatives of the rotating eigenvectors.

Qualification of the coordinate claim. The displayed equation for ${\sigma }^v$ is not true for arbitrary coordinate displacements along arbitrary reference geodesics. For synchronized neighboring curves with $u=\tau$ and ${\sigma }^u=0$, the exact geodesic equations areTheir variation givesThe second expression need not vanish. A concrete vacuum counterexample is $M=\mathrm{diag}(1,-1)$ and the one-parameter family of timelike geodesicsEach has $g(V,V)=-1$. Differentiating with respect to $A$ produces a Jacobi fieldThis even has $g(V,\sigma )=0$, so it is a physical orthogonal separation. The covariant derivative form of geodesic deviation remains correct; the unrestricted assertion about ordinary coordinate derivatives needs the adapted reference and frame used above. If instead $\dot{u}=c$ is a constant affine parameter normalization, the transverse tidal coefficient is $c^{2}M(u)$.

For the plane-fronted gravitational wave $d{s}^2=2dudv+d{\mathbf{x}}^2+{\mathbf{x}}^{T}M(u)\mathbf{x}d{u}^2$, synchronized neighboring geodesics parametrized by $u$ have transverse separation $\mathbf{\xi }$ satisfying the displayed equation. Along the central timelike geodesic $(u,v,\mathbf{x})=(\tau ,-\tau /2,0)$ the coordinate basis is parallel transport, so this is also the transverse Jacobi field equation in a transported frame. Vacuum requires $\mathrm{tr}M=0$, producing equal and opposite instantaneous transverse tidal acceleration eigenvalues. The ordinary coordinate equation ${\ddot{\xi }}^v=0$ holds on that central reference, but generally fails on off-axis reference curves.